Therefore, in the local case, the Hamiltonian function \(\mathcal {H}\)  (44) in [1]} and \(H\)  in [2]} satisfy the following relation:
\(\begin{aligned}&\mathcal {H}_u(t,x,y,z,u,\tilde{p},\tilde{q})=\frac{1}{q}H_u(t,x,y,z,p,k,q,u).\end{aligned}\) 
